Page 219 - ICSE Math 8

Tangents

A tangent is a line that touches a circle at only one point. The point at which a
tangent meets a circle is known as its point of contact. In the adjoining figure,
PQ is a tangent to the circle with R as its point of contact. At the point of contact,
radius and tangent are perpendicular to each other.

Points to remember

• One and only one tangent can be drawn through a point on the circle.
• From a point outside the circle, two tangents can be drawn and both these
tangents are always equal to each other.
In the adjoining figure, two tangents AB and AC are drawn from point A. Thus,
AB = AC.

Secant
A secant is a line that intersects a circle at two distinct points. In the adjoining figure, XY is a secant intersecting
the circle at point A and B.

Point to remember

A secant is a line passing through the circle intersecting it at two distinct points,
whereas a chord is a line segment joining any two points on the circle. In the
adjoining figure, XY is a secant whereas PQ is a chord.

Example 4: In the given figure, PQ and PR are two tangents to the circle of radius 3 cm from external
point P. If PQ = 4 cm, find OP and PR.
Solution: OQ ⊥ PQ ( Radius is perpendicular to the tangent at the point of contact)
∴ In right-angled ∆OQP,
2
2
2
OP = OQ + PQ (Using Pythagoras theorem)
2
2
2
⇒ OP = 3 + 4 = 9 + 16 = 25 ⇒ OP = 25 = 5 cm
Also, PR = PQ ( Tangents from an external point are equal)
∴ PR = 4 cm
Example 5: C and C are two concentric circles. Chord PQ of circle C touches circle C at A. Show that
1
1
2
2
OA bisects PQ.
Solution: Chord PQ of C is a tangent to circle C at A.
1
2
∴ OA ⊥ PQ ( Radius is perpendicular to the tangent at the point of contact)
∴ In right-angled ∆OAP and right-angled ∆OAQ,
OP = OQ (Radii of circle C )
2
OA = OA (Common)
∠OAP = ∠OAQ (Both 90°)
∴ ∆OAP ≅ ∆OAQ (By RHS condition)
Hence, AP = AQ (Corresponding sides of congruent triangles are equal)

• A line from the centre of a circle perpendicular to the chord bisects the chord.
• Conversely, a line from the centre of a circle bisecting the given chord is perpendicular to the chord.

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